Hi, i play with a my friend in Lan Mode. But when he try to open the Tardis it says: "No permission to enter". And when i open the door for him. He left the game. What is the problem?

The Tardis permissions system operates on UUIDs of players and checks if your UUID matches one that is allowed to open the Tardis door.

This is only available on an online server. LAN is essentially offline mode and therefore has no way of checking your UUID. Therefore when you try to open the door, the server gets confused and can kick players such as your friend.

You'll need to be on a server for your friend to allow you in.

Once you are on a server, your friend (the owner of the Tardis) must type:

/tardis-companions add [Your username]

This command allows you be a companion of a Tardis and will allow you to enter it if the door has been opened already.

Your friend must then type:

/tardis-perms companions

This command allows any registered companion (you) to open the Tardis door and fly it.

Thanks! 

Is the server only Premium? Until this summer I will not be able to buy a premium account. So I was wondering if I'll have to wait or not.

Yes. If the server wasn't premium then the mod would break and so would all ranks and players.

There's a thread on this [Reference] "Will you make the server cracked".